Question: Simplify; express your answer in exponential form. Assume $t\neq 0, p\neq 0$. $\dfrac{{(t^{-5}p^{4})^{-1}}}{{(t^{5}p^{-1})^{2}}}$
Solution: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(t^{-5}p^{4})^{-1} = (t^{-5})^{-1}(p^{4})^{-1}}$ On the left, we have ${t^{-5}}$ to the exponent ${-1}$ . Now ${-5 \times -1 = 5}$ , so ${(t^{-5})^{-1} = t^{5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(t^{-5}p^{4})^{-1}}}{{(t^{5}p^{-1})^{2}}} = \dfrac{{t^{5}p^{-4}}}{{t^{10}p^{-2}}}$ Break up the equation by variable and simplify. $\dfrac{{t^{5}p^{-4}}}{{t^{10}p^{-2}}} = \dfrac{{t^{5}}}{{t^{10}}} \cdot \dfrac{{p^{-4}}}{{p^{-2}}} = t^{{5} - {10}} \cdot p^{{-4} - {(-2)}} = t^{-5}p^{-2}$